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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写力学 mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写力学 mechanics代写方面经验极为丰富，各种代写力学 mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|力学代写mechanics代考|GENERAL INTERPRETATION OF PRINCIPAL STRESSES

In the following we will consider three geometric descriptions of the principal stresses.

If all three principal stresses are equal, no shear stresses will exist. This means that the principal stresses exist in all directions. If the stress state is plotted in space, it can be presented as a sphere. This is shown in Figure 3.2, and is known as hydrostatic state of stress.

A more complex case arises when two of the principal stresses are equal, but different from the third. There will be symmetry in the plane which is orthogonal to the third stress component. A geometric presentation of this stress state is shown in Figure 3.3.

When rock core plugs are tested in the laboratory, the cylindrical stress state of Figure $3.3$ is normally used. Later we will show that this stress state is also often used for wellbore instability analysis.

The third stress state is the so-called triaxial stress state. In this case all principal stresses have different magnitudes, as shown in Figure 3.4.

## 物理代写|力学代写mechanics代考|TWO-DIMENSIONAL STRESS ANALYSIS

Considering a two-dimensional loading case, where there is no stress along the $z$-axis, we would like to find the principal stresses. For this case, the stresses are $\sigma_{z}=\tau_{x z}=\tau_{y z}=0$, and Equation $1.3$ is simplified to:
$$[\boldsymbol{\sigma}]=\left[\begin{array}{ccc} \sigma_{x} & \tau_{x y} & 0 \ \tau_{x y} & \sigma_{y} & 0 \ 0 & 0 & 0 \end{array}\right]=\left[\begin{array}{ll} \sigma_{x} & \tau_{x y} \ \tau_{x y} & \sigma_{y} \end{array}\right]$$
The invariants of Equation $3.4$ can also be simplified to:
\begin{aligned} &I_{1}=\sigma_{x}+\sigma_{y} \ &I_{2}=\tau_{x y}^{2}-\sigma_{x} \sigma_{\gamma} \ &I_{3}=0 \end{aligned}
and the equation for the principal stresses becomes:
$$\sigma\left(\sigma^{2}-I_{1} \sigma-I_{2}\right)=0$$
or
$$\sigma^{2}-\left(\sigma_{x}+\sigma_{y}\right) \sigma-\left(\tau_{x y}^{2}-\sigma_{x} \sigma_{y}\right)=0$$
which is a quadratic equation. This equation has two roots, which are given by:
$$\sigma_{1,2}=\frac{1}{2}\left(\sigma_{x}+\sigma_{\gamma}\right) \pm \sqrt{\left(\frac{1}{2}\left(\sigma_{x}-\sigma_{\gamma}\right)^{2}\right)+\tau_{x y}^{2}}$$
where $\sigma_{1}$ and $\sigma_{2}$ are the maximum and minimum principal stresses. Equation $3.10$ in fact represents the equation for Mohr’s circle, as shown in Figure 3.5.

# 力学代考

## 物理代写|力学代写mechanics代考|TWO-DIMENSIONAL STRESS ANALYSIS

$$I_{1}=\sigma_{x}+\sigma_{y} \quad I_{2}=\tau_{x y}^{2}-\sigma_{x} \sigma_{\gamma} I_{3}=0$$

$$\sigma\left(\sigma^{2}-I_{1} \sigma-I_{2}\right)=0$$

$$\sigma^{2}-\left(\sigma_{x}+\sigma_{y}\right) \sigma-\left(\tau_{x y}^{2}-\sigma_{x} \sigma_{y}\right)=0$$

$$\sigma_{1,2}=\frac{1}{2}\left(\sigma_{x}+\sigma_{\gamma}\right) \pm \sqrt{\left(\frac{1}{2}\left(\sigma_{x}-\sigma_{\gamma}\right)^{2}\right)+\tau_{x y}^{2}}$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师